Question

figure q was the result of a sequence of transformations on figure p, both shown below.

Explanation:

Since the problem is about identifying the transformations from figure P to figure Q, and it's a geometry (subfield of Mathematics) problem involving coordinate plane transformations, we can analyze the steps:

Step1: Analyze Horizontal Translation

First, observe the horizontal shift. The center of figure P (e.g., the midpoint of its base) is at \( x = -3.5 \) (midpoint of -5 and -2). The center of figure Q's base is at \( x = 3.5 \) (midpoint of 2 and 5? Wait, Q's base is from x=2 to x=5? Wait, looking at the grid, P is from x=-5 to x=-2 (width 3 units), Q is from x=2 to x=5 (width 3 units). So horizontal shift: from x=-3.5 to x=3.5, that's a shift of \( 3.5 - (-3.5) = 7 \) units right? Wait, or maybe count the distance between corresponding points. Let's take a vertex of P, say (-5, 0). The corresponding vertex in Q: let's see Q's base is on y=0? Wait no, Q's base is on y=0? Wait P's base is on y=0 (from (-5,0) to (-2,0)), and Q's base is on y=0 (from (2,0) to (5,0))? Wait no, looking at the graph, P is above the x-axis, Q is below? Wait P's bottom is on y=0, top of the curve is at y=2? Wait no, P's figure: the vertical sides are from (-5,0) to (-5,3), (-2,0) to (-2,3), and the curve is from (-5,3) to (-2,3) curving down to y=1? Wait maybe better to check the transformation: reflection and translation.

Step2: Analyze Reflection

Figure P is above the x-axis, figure Q is below the x-axis, so there's a reflection over the x-axis (since the y-coordinates are negated). Then, horizontal translation: from x=-5 to x=2 (a point on P: (-5,0) to (2,0) on Q? Wait (-5,0) to (2,0): that's a shift of 7 units right? Wait 2 - (-5) = 7. Or from -3.5 (midpoint of -5 and -2) to 3.5 (midpoint of 2 and 5), which is 7 units right.

So the sequence of transformations could be: 1. Reflect figure P over the x - axis (to flip it vertically over the x - axis, changing the orientation of the curve from upward - opening to downward - opening). 2. Translate the reflected figure 7 units to the right (to move it from the left side of the y - axis to the right side, aligning with figure Q's position).

Answer:

The sequence of transformations from figure P to figure Q is a reflection over the x - axis followed by a horizontal translation of 7 units to the right (or other valid sequences like translation then reflection, but reflection over x - axis and translation right is a common one).